Warm Up 1. What is the name of the point where the angle bisectors of a triangle intersect? incenter Find the midpoint of the segment with the given endpoints. 2. (1, 6) and (3, 0) 3. (7, 2) and (3, 8) 4. Write an equation of the line containing the
(1, 3) slope form. points (3, 1) and (2, 10) in point- (5, 3) y 1 = 9(x 3)
Objectives Apply properties of medians of a triangle. Apply properties of altitudes of a triangle. Vocabulary median of a triangle centroid of a triangle altitude of a triangle orthocenter of a triangle
A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Every triangle has three medians, and the medians are concurrent. The point of concurrency of the medians of a triangle is the centroid of the triangle . The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance.
Example 1A: Using the Centroid to Find Segment Lengths In LMN, RL = 21 and SQ =4. Find LS. Centroid Thm. Substitute 21 for RL. LS = 14 Simplify.
Example 1B: Using the Centroid to Find Segment Lengths In LMN, RL = 21 and SQ =4. Find NQ. Centroid Thm. NS + SQ = NQ Seg. Add. Post. Substitute
Subtract NQ for NS. from both sides. Substitute 4 for SQ. 12 = NQ Multiply both sides by 3.
Check It Out! Example 1a In JKL, ZW = 7, and LX = 8.1. Find KW. Centroid Thm. Substitute 7 for ZW. KW = 21 Multiply both sides by 3.
Check It Out! Example 1b In JKL, ZW = 7, and LX = 8.1. Find LZ. Centroid Thm. Substitute 8.1 for LX. LZ = 5.4 Simplify.
Example 2: Problem-Solving Application A sculptor is shaping a triangular piece of iron that will balance on the point of a cone. At what coordinates will the triangular region balance? Example 2 Continued Understand the Problem
1 The answer will be the coordinates of the centroid of the triangle. The important information is the location of the vertices, A(6, 6), B(10, 7), and C(8, 2). 2
Make a Plan The centroid of the triangle is the point of intersection of the three medians. So write the equations for two medians and find their point of intersection. Example 2 Continued 3 Solve
Let M be the midpoint of AB and N be the midpoint of AC. CM is vertical. Its equation is x = 8. BN has slope 1. Its equation is y = x 3. The coordinates of the centroid are D(8, 5). Example 2 Continued 4 Look Back
Let L be the midpoint of BC. The equation for AL is 8 at D(8, 5). , which intersects x = Check It Out! Example 2 Find the average of the x-coordinates and the average of the y-coordinates of the vertices of PQR. Make a conjecture about the centroid of a triangle.
Check It Out! Example 2 Continued The x-coordinates are 0, 6 and 3. The average is 3. The y-coordinates are 8, 4 and 0. The average is 4. The x-coordinate of the centroid is the average of the x-coordinates of the vertices of the , and the y-coordinate of the centroid is the
average of the y-coordinates of the vertices of the . An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle. In QRS, altitude QY is inside the triangle, but RX and SZ are not. Notice that the lines containing the altitudes are concurrent at P. This point of concurrency is the orthocenter
of the triangle. Helpful Hint The height of a triangle is the length of an altitude. Example 3: Finding the Orthocenter Find the orthocenter of XYZ with vertices X(3, 2), Y(3, 6), and Z(7, 1).
Step 1 Graph the triangle. X Example 3 Continued Step 2 Find an equation of the line containing the altitude from Z to XY. Since XY is vertical, the altitude is horizontal. The line containing it must pass through Z(7, 1) so the equation of the line is y = 1.
Example 3 Continued Step 3 Find an equation of the line containing the altitude from Y to XZ. The slope of a line perpendicular to XZ is . This line must pass through Y(3, 6). Point-slope form. Substitute 6 for y1,
and 3 for x1. Distribute . Add 6 to both sides. for m,
Example 3 Continued Step 4 Solve the system to find the coordinates of the orthocenter. Substitute 1 for y. Subtract 10 from both sides. 6.75 = x Multiply both sides by The coordinates of the orthocenter are (6.75, 1).
Check It Out! Example 3 Show that the altitude to JK passes through the orthocenter of JKL. An equation of the altitude to JK is 4=1+3 4=4 Therefore, this altitude passes through the orthocenter.
Lesson Quiz Use the figure for Items 13. In ABC, AE = 12, DG = 7, and BG = 9. Find each length. 1. AG 8 2. GC 14 3. GF 13.5 For Items 4 and 5, use MNP with vertices M (4, 2), N (6, 2) , and P (2, 10). Find the
coordinates of each point. 4. the centroid (0, 2) 5. the orthocenter
Draw a circuit diagram of a closed series circuit with a 1.5V battery and three light bulbs. Series Circuits. Series Circuits. What happens if there is a break anywhere in a series circuit? Series Circuits. Why are Christmas lights wired...
Once the irrigant is suctioned out, chest tubes of the surgeon's choice will be placed using a 10 blade on a #3 knife handle (incisions are made below the thoracotomy incision), cautery may be used, a tonsil or kelly will...
Therefore, a total of 5,400 ft.2 would be required for a full kitchen food service facility. Restrooms Other Methods to Determine Space Requirements Converting Method The present space requirements are converted to those required for the proposed layout.
Shows increased productivity and competitiveness of EU agriculture and food industry. Underlines need to explore and develop trade relations with third countries. EU in global agri-food trade. Value of EU trade in agri-food products (exports and imports) at EUR 255.3...
Weld Metal Protection. During fusion welding, the molten metal in the weld "puddle" is susceptible to oxidation. Must protect weld puddle (arc pool) from the atmosphere. Methods. Weld Fluxes. Inert Gases. Vacuum
Distributed Science Methodology publishes all steps in a new electronic logbook capturing scientific process (data analysis) as a rich cloud of resources including emails, PPT, Wikis as well as databases, compiler options, build time/runtime configuration… Community (?
Ready to download the document? Go ahead and hit continue!